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High-Order Compact Finite-Difference Scheme for Singularly-Perturbed Reaction-Diffusion Problems on a New Mesh of Shishkin Type

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  • V. Kumar

    (Ecole Normale Superieure)

Abstract

In this work, a high-order compact finite-difference (HOCFD) scheme has been proposed to solve 1-dimensional (1D) and 2-dimensional (2D) elliptic and parabolic singularly-perturbed reaction-diffusion problems. A new kind of piecewise uniform mesh of Shishkin type (Miller et al. in Fitted Numerical Methods for Singular Perturbation Problems, 1996) has also been proposed and using this mesh the HOCFD scheme gives better results as compared to the results using the Shishkin mesh. Moreover, the stated method gives ε-uniform convergence and improved orders of convergence which have also been provided in the results for some test problems.

Suggested Citation

  • V. Kumar, 2009. "High-Order Compact Finite-Difference Scheme for Singularly-Perturbed Reaction-Diffusion Problems on a New Mesh of Shishkin Type," Journal of Optimization Theory and Applications, Springer, vol. 143(1), pages 123-147, October.
    • Handle: RePEc:spr:joptap:v:143:y:2009:i:1:d:10.1007_s10957-009-9547-y
    DOI: 10.1007/s10957-009-9547-y
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    Cited by:

    1. Anindya Goswami & Kuldip Singh Patel & Pradeep Kumar Sahu, 2022. "A novel difference equation approach for the stability and robustness of compact schemes for variable coefficient PDEs," Papers 2209.02873, arXiv.org, revised Jan 2024.
    2. Chen, Changkai & Zhang, Xiaohua & Liu, Zhang, 2020. "A high-order compact finite difference scheme and precise integration method based on modified Hopf-Cole transformation for numerical simulation of n-dimensional Burgers’ system," Applied Mathematics and Computation, Elsevier, vol. 372(C).
    3. Anindya Goswami & Kuldip Singh Patel, 2022. "Matrix method stability and robustness of compact schemes for parabolic PDEs," Papers 2201.05854, arXiv.org, revised Jan 2024.

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